Solving Equations Examples And Worksheet Page 2
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14A50
Appendix A
Review of Fundamental Concepts of Algebra
A linear equation in one variable, written in standard form, always has exactly one
solution. To see this, consider the following steps.
Note that some linear equations
ax
b
0
Original equation, with
a
0
in nonstandard form have no
ax
b
Subtract b from each side.
solution or infinitely many
solutions. For instance,
b
x
Divide each side by a.
a
x
x
1
To solve a conditional equation in
x,
isolate
x
on one side of the equation
has no solution because it is not
by a sequence of equivalent (and usually simpler) equations, each having the same
true for any value of
x.
Because
solution(s) as the original equation. The operations that yield equivalent equations come
5x
10
5 x
2
from the Substitution Principle (see Appendix A.1) and simplification techniques.
is true for any value of
x,
the
equation has infinitely many
Generating Equivalent Equations
solutions.
An equation can be transformed into an equivalent equation by one or more of
the following steps.
Equivalent
Given Equation
Equation
1. Remove symbols of grouping,
2x
x
4
x
4
combine like terms, or simplify
fractions on one or both sides
of the equation.
2. Add (or subtract) the same
x
1
6
x
5
quantity to (from) each side
of the equation.
3. Multiply (or divide) each
2x
6
x
3
side of the equation by the
same nonzero quantity.
4. Interchange the two sides of
2
x
x
2
the equation.
Example 1
Solving a Linear Equation
After solving an equation, you
should check each solution in the
3x
6
0
a.
Original equation
original equation. For instance,
3x
6
Add 6 to each side.
you can check the solution of
Example 1(a) as follows.
x
2
Divide each side by 3.
Write original
3x
6
0
5x
4
3x
8
b.
Original equation
equation.
?
3
2
6
0
Substitute 2 for x.
2x
4
8
Subtract
3x
from each side.
✓
0
0
Solution checks.
2x
12
Subtract 4 from each side.
Try checking the solution of
x
6
Divide each side by 2.
Example 1(b).
Now try Exercise 15.
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